Theorem 2.5 (Theorem G, [1]}) 
Let \(f\)  be an orientation-preserving homeomorphism of \(\mathbb {S}^2\)  that has no topological horseshoe. Then every non-wandering point \(z\)  whose \(\alpha \) -limit set or \(\omega \) -limit set in not included in the set of fixed points of \(f\)  lies in an open annulus \(A\)  which is a fixed point free and \(f\vert _{A}\)  is isotopic to the identity.

